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summer house on the lake - Deep Underground Poetry

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 The magnitude of the perpendicular component (which is the desired quantity for speed component):

📌 \|\mathbf{v}_{\perp}\| = \sqrt{(2.5)^2 + 0^2 + (2.5)^2} = \sqrt{6.25 + 6.25} = \sqrt{12.5} = rac{5\sqrt{2}}{2}

📌 Alternatively, using formula for perpendicular component magnitude:

📌 \|\mathbf{v}_{\perp}\| = \sqrt{\|\mathbf{v}\|^2 - \left(rac{|\mathbf{v} \cdot \mathbf{c}|}{\|\mathbf{c}\|}

📌 But since we already computed it:

📌 oxed{\dfrac{5\sqrt{2}}{2}}

📌 A materials scientist is studying the symmetry of molecular arrangements in a self-healing material. The arrangement has a rotational symmetry group isomorphic to the dihedral group \(D_4\). How many distinct vectors \(\mathbf{v} = egin{pmatrix} x \ y \end{pmatrix}\) in \(\mathbb{R}^2\) with integer components satisfy \( \mathbf{v} \cdot \mathbf{v} = 10 \) and are invariant under a \(90^\circ\) rotation about the origin?

📌 The standard \(90^\circ\) rotation matrix is:

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